Key Concepts

#Steps to Find the Constant

1. Integrate the Function:
   • Perform the integration to find the general form of the antiderivative.
2. Use Additional Information:
   • If additional information is provided, such as a specific value of $F(x)$ for a given $x$, or a point $(x_0, y_0)$ that the graph passes through, use this to find $C$.
3. Set Up and Solve an Equation:
   • Use the given point to set up an equation and solve for $C$.

#Example Process

Suppose we need to integrate $f(x) = 2x + 3$ and find the constant of integration given that $F(x)$ goes through the point $(1, 2)$.

1. Integrate: $$F(x) = \int (2x + 3) , dx = x^2 + 3x + C$$

2. Use Given Point:

   • Given $F(1) = 2$, substitute $x = 1$ and $F(x) = 2$ into the equation: $$2 = 1^2 + 3(1) + C$$

3. Solve for $C$: $$2 = 1 + 3 + C \Rightarrow 2 = 4 + C \Rightarrow C = -2$$

Thus, the specific antiderivative is: $$F(x) = x^2 + 3x - 2$$

#Worked Example

The function $h$ measures the height of a projectile above the ground at time $t$ (in seconds). It is known that $h$ satisfies the equation:

$$h(t) = \int (70 - 32t) , dt$$

and at time $t = 2$, the projectile is 81 feet above the ground.

Find an explicit expression for $h$ in terms of $t$.

#Solution

1. Integrate: $$h(t) = \int (70 - 32t) , dt = 70t - 16t^2 + C$$

2. Use Given Point:

   • Given $h(2) = 81$, substitute $t = 2$ and $h(t) = 81$ into the equation: $$81 = 70(2) - 16(2)^2 + C$$

3. Solve for $C$: $$81 = 140 - 64 + C \Rightarrow 81 = 76 + C \Rightarrow C = 5$$

Thus, the explicit expression for $h$ is: $$h(t) = 70t - 16t^2 + 5$$

#Practice Questions

Practice Question

1. Find the constant of integration for the integral $\int (4x - 7) , dx$ given that the antiderivative passes through the point $(2, 1)$.

Practice Question

2. Integrate the function $f(x) = 3x^2 - 2x + 1$ and determine the constant of integration if the function passes through the point $(1, 4)$.

#Glossary

• Constant of Integration ($C$): An arbitrary constant added to the antiderivative of a function when computing an indefinite integral.
• Indefinite Integral: An integral without upper and lower limits, representing a family of functions.
• Antiderivative: A function whose derivative is the given function.

#Summary and Key Takeaways

• When finding an indefinite integral, always include the constant of integration, $C$.
• Use additional information, such as specific points or values, to solve for $C$.
• The process of finding $C$ is equivalent to finding the particular solution to a differential equation.

#Exam Strategy

• Read the Problem Carefully: Ensure you understand all given conditions and points.
• Double-Check Integration: Make sure your integration is correct and includes $C$.
• Use Given Points Accurately: Substitute them correctly into the integrated function to find $C$.

By following these strategies and understanding the process, you'll be better prepared to handle questions involving constants of integration in your exams.